Notes on: “Back of the envelope calculations in friction stir welding - velocities, peak temperature,

torque, and hardness”

A. Arora, H. K. D. H. Bhadeshia and T. DebRoy

Contact email: debroy@psu.edu

Analytical calculations of the material flow fields, peak temperature, torque, and

weld hardness for friction stir welding are available in the following paper:

A. Arora, T. DebRoy, H.K.D.H. Bhadeshia, Back-of-the-envelope calculations in friction stir

welding – Velocities, peak temperature, torque, and hardness, Acta Materialia 59(2011) 2020-

2028. doi:10.1016/j.actamat.2010.12.001

In order to facilitate these calculations, links to several downloadable files are

provided at the end of this document. Both the source code and the executable file for the

calculation of velocities are available for download. For the calculation of peak

temperature, torque and weld hardness excel spreadsheets are provided. The following

notes give descriptions of the files and the calculation procedures.

Velocity field Velocity field during friction stir welding (FSW) can be calculated using the

accompanying executable file named velocity.exe or the Fortran source code file named

velocity.for. The executable file can be used on any Microsoft Windows® operating

system. Upon execution the user will be asked to select the workpiece material and

values of a few welding parameters. The source code can be compiled using any standard

Fortran compiler. The source code has two subroutines named grid and fgh.

The subroutine grid creates a computational domain with length (x) and width (y)

equal to the tool shoulder radius (rs) and depth (z) equal to the pin length (pl). The length

and width of computational domain is divided into 100 equal divisions each and depth is

divided into 40 divisions. An inverted cone shaped computational domain defined by

rh(k) is considered with radius at the top surface equal to the tool shoulder radius and at

the bottom equal to the pin radius (rp).

The subroutine fgh computes the values of function F, G and H which are then

used to compute the velocity components ut, vt, wx in the r, θ and z directions in

cylindrical coordinate system. These velocities are then converted to cartesian coordinate

system (ux, vx and wx). The velocities are written to a file entitled “output.txt” for

plotting using a commercial software such as Tecplot®. A detailed description of the

equations used can be obtained in the manuscript entitled “Back of the envelope

calculations in friction stir welding - velocities, peak temperature, torque, and hardness”.

Using the default variable values in the program velocity.exe the following stream

traces were generated. The stream traces show that the computed velocities display the

flow asymmetry characteristic of FSW where most of the material flows through the

retreating side. The analytically computed velocity fields can be further used to compute

strain and strain rates in FSW.

0

5

10

15

20

25

30

Y

0 5 10 15 20 25 30

X

200 mm/s 0.16 mm awayfrom shoulder

,mm

, mm

0

5

10

15

20

25

30

Y0 5 10 15 20 25 30

X

200 mm/s 2.3 mm awayfrom shoulder

,mm

, mm Figure 1 Analytically computed velocity fields and stream traces in FSW at two planes parallel to the tool shoulder workpiece interface. The planes are 0.16 and 2.3 mm away from the tool shoulder.

Peak Temperature, Torque and TMAZ hardness: Three spreadsheets are provided to calculate the peak temperature, torque and

TMAZ hardness in FSW. The spreadsheet entitled “Peak temperature.xls” can be used to

calculate the peak temperature in friction stir welding. The data used for the derivation of

dimensionless correlation are available in the spreadsheet. The variable names and units

are mentioned in the first cell of every column. Further details about the estimation of

peak temperature in FSW can be found in the manuscript entitled “Back of the envelope

calculations in friction stir welding - velocities, peak temperature, torque, and hardness”.

The torque in FSW can be analytically calculated using spreadsheet entitled

“Torque.xls”. The calculated torque values for AA2524 and Ti-6A-4V are provided in

the spreadsheet. The variables used with the units are shown on the top of every column.

For the formulas and description of methodology please refer to the manuscript entitled

“Back of the envelope calculations in friction stir welding - velocities, peak temperature,

torque, and hardness”.

The TMAZ hardness in friction stir welding of aluminum alloys can be estimated

analytically as a function of the concentration of the alloying elements. The analytically

estimated hardness values for various aluminum alloys is shown in spreadsheet named

“Aluminum Hardness.xls”. The reference numbers given in the spreadsheet correspond to

the references in the manuscript entitled “Back of the envelope calculations in friction stir

welding - velocities, peak temperature, torque, and hardness”.

List of files accompanying this document: 1. velocity.exe 2. velocity.for 3. Peak temperature.xls 4. Torque.xls 5. Aluminum Hardness.xls